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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclscls00 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.) |
Ref | Expression |
---|---|
ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclscls00 | ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsfv1 41665 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
5 | 4 | fveq1d 6776 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅)) |
6 | 2, 3 | ntrclsbex 41644 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | 1, 2, 3 | ntrclsiex 41663 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
8 | eqid 2738 | . . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼) | |
9 | 0elpw 5278 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵 | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵) |
11 | eqid 2738 | . . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅) | |
12 | 1, 2, 6, 7, 8, 10, 11 | dssmapfv3d 41627 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
13 | 5, 12 | eqtr3d 2780 | . . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
14 | dif0 4306 | . . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵 | |
15 | 14 | fveq2i 6777 | . . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵) |
16 | id 22 | . . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵) | |
17 | 15, 16 | eqtrid 2790 | . . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵) |
18 | 17 | difeq2d 4057 | . . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵)) |
19 | difid 4304 | . . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
20 | 18, 19 | eqtrdi 2794 | . . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅) |
21 | 13, 20 | sylan9eq 2798 | . 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅) |
22 | pwidg 4555 | . . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵) | |
23 | 6, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
24 | 1, 2, 3, 23 | ntrclsfv 41669 | . . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵)))) |
25 | 19 | fveq2i 6777 | . . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅) |
26 | id 22 | . . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅) | |
27 | 25, 26 | eqtrid 2790 | . . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅) |
28 | 27 | difeq2d 4057 | . . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅)) |
29 | 28, 14 | eqtrdi 2794 | . . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵) |
30 | 24, 29 | sylan9eq 2798 | . 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵) |
31 | 21, 30 | impbida 798 | 1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∖ cdif 3884 ∅c0 4256 𝒫 cpw 4533 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ↑m cmap 8615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 |
This theorem is referenced by: (None) |
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